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TL;DR Place a converging (convex) lens between an illuminated object and a white screen. For several object distances u, find the image distance v that gives a sharp image, then calculate 1/u and 1/v. Plot 1/v against 1/u
The focal lengthf of a converging lens is the distance from the optical centre of the lens to the principal focus (focal point).
When a beam of light rays arrives parallel to the principal axis of a converging (convex) lens, the lens refracts the rays so that they all pass through a single point on the other side. That point is the principal focus, and the distance from the centre of the lens to this point is the focal length.
A shorter focal length means the lens is more powerful (it bends light more strongly). The power of a lens is defined as:
P=f1
where P is in dioptres (D) when f is in metres.
For this experiment, you will determine f in centimetres since all distances on an optical bench are measured in cm.
This is the simplest way to find the focal length and makes a good starting check before the full experiment.
Procedure
Hold the converging lens and point it at a distant, well-lit object such as a window, a tree, or a building across the field. The object should be at least 10 metres away.
Hold a white screen (or a sheet of paper) behind the lens.
Move the screen back and forth until a sharp, inverted image of the distant object appears on it.
Measure the distance from the centre of the lens to the screen. This distance is approximately equal to the focal length f.
Why this works
Rays from a very distant object arrive at the lens approximately parallel to each other. When parallel rays pass through a converging lens, they converge at the principal focus. So the image forms at the focal point, and the lens-to-screen distance equals f.
Strictly, the thin-lens equation gives:
f1=u1+v1
When u is very large, 1/u≈0, so v≈f.
This method is fast but only an approximation. For a more precise result, use the lens equation method below.
This method collects multiple data points and uses a graph to determine f more reliably.
The thin-lens equation
For a thin converging lens forming a real image:
f1=u1+v1
where:
u = distance from the object to the centre of the lens (object distance), in cm
v = distance from the centre of the lens to the screen (image distance), in cm
f = focal length of the lens, in cm
Rearranging for 1/v:
v1=f1−u1=−u1+f1
This has the form y=mx+c. If you plot 1/v on the y-axis against 1/u on the x-axis, you get a straight line with:
Gradient=−1
y-intercept=1/f
x-intercept=1/f (set 1/v=0)
So both the y-intercept and the x-intercept give 1/f, and you can calculate f from either one.
4 | Apparatus
Item
Purpose
Converging (convex) lens
The lens whose focal length you are finding
Lens holder
Holds the lens upright and perpendicular to the bench
Illuminated object (crosswire slide or wire gauze with lamp)
Provides a well-defined object whose image can be focused sharply
White screen
The surface on which the real image forms
Metre ruler or optical bench with scale
Measures object distance u and image distance v
Screen holder / retort stand and clamp
Supports the screen at the same height as the lens centre
If an optical bench is available, use it. The built-in scale and sliding mounts make it much easier to measure u and v accurately and to keep everything aligned along the principal axis.
5 | Step-by-Step Method (Lens Equation Approach)
Set up the apparatus. Place the illuminated object at one end of the optical bench (or metre ruler). Mount the converging lens in its holder and position it along the bench. Place the white screen on the other side of the lens.
Estimate f first. Use the distant object method (Section 2) to get a rough value of the focal length. This tells you the minimum object distance to use --- all values of u must be greater than f to produce a real image.
Set the first object distance. Move the lens so that u (the distance from the illuminated object to the centre of the lens) is well above f. A good starting value is roughly 2f.
Find the sharp image. Slide the screen back and forth until a sharp, well-focused image of the crosswire appears on the screen. Record the image distance v (centre of the lens to the screen).
Repeat for at least 6 values of u. Change u each time, ensuring all values are greater than f. Spread your values of u across a wide range --- do not cluster them. For each new u, re-adjust the screen to find the sharp image and record v.
Calculate 1/u and 1/v for each pair of readings. Enter these in your data table.
Practical tips:
Darken the room so the image on the screen is easier to see.
Check that the lens, object, and screen centres are all at the same height.
Measure u and v from the centre of the lens, not from the edge or the holder.
For each u, find the sharp image, then overshoot and come back to confirm the best position. This reduces the uncertainty from depth of focus.
6 | Data Table Template
Record at least six rows. All columns must include the quantity and unit separated by a forward slash.
Reading
u / cm
v / cm
1/u / cm−1
1/v / cm−1
1
2
3
4
5
6
Record u and v to the nearest 0.1 cm if using an optical bench, or to the nearest 0.5 cm if using a metre ruler.
Calculate 1/u and 1/v to 3 or 4 significant figures.
Check: if any value of v comes out negative or extremely large, you may have set u<f (no real image forms).
7 | Plotting the Graph
Axes
x-axis:1/u / cm−1
y-axis:1/v / cm−1
What to do
Choose suitable scales so the plotted points spread across at least half of each axis. Do not start either axis at zero unless the data naturally includes values near zero.
Plot all six (or more) data points.
Draw the best-fit straight line through the points. The line should have a negative gradient (sloping downward from left to right).
Extend the best-fit line to meet both axes.
Reading the focal length from the graph
The y-intercept (where the line meets the y-axis, i.e. 1/u=0) equals 1/f.
The x-intercept (where the line meets the x-axis, i.e. 1/v=0) also equals 1/f.
In theory, both intercepts should give the same value. In practice, they may differ slightly because of experimental scatter. If they do differ, average the two values of f obtained from each intercept.
Suppose you obtain the following data for a converging lens:
Reading
u / cm
v / cm
1/u / cm−1
1/v / cm−1
1
20.0
20.0
0.0500
0.0500
2
25.0
16.7
0.0400
0.0599
3
30.0
15.0
0.0333
0.0667
4
35.0
14.0
0.0286
0.0714
5
40.0
13.3
0.0250
0.0752
6
50.0
12.5
0.0200
0.0800
From the graph
After plotting 1/v (y-axis) against 1/u (x-axis), the best-fit line has:
y-intercept≈0.100 cm−1
x-intercept≈0.100 cm−1
Since both intercepts equal 1/f:
f1=0.100cm−1
f=0.1001=10.0cm
Quick check with one data point
Using reading 1 (u=20.0 cm, v=20.0 cm):
f1=20.01+20.01=0.0500+0.0500=0.100cm−1
f=10.0cm
This confirms the graph result. Note that when u=v=2f, the object and image distances are equal. This is a useful check --- if you find a position where u=v, then f=u/2.
9 | Sources of Error
Thick lens approximation. The thin-lens equation assumes the lens has negligible thickness. A thick lens has two principal planes, and u and v should strictly be measured from these planes rather than from the physical centre of the lens. This causes a small systematic error in every reading.
Parallax error when measuring distances. If your eye is not level with the ruler markings when reading u or v, the recorded values will be slightly off. Always read the ruler at eye level, perpendicular to the scale.
Judging the sharpest image. There is a range of screen positions over which the image appears approximately sharp (depth of focus). This makes it difficult to pinpoint the exact position of the sharp image, introducing random error in v. To reduce this, overshoot the sharp position in both directions and take the midpoint.
Lens not perpendicular to the bench. If the lens is tilted, the principal axis does not run along the bench. The image may be distorted and the measured distances will not correspond to the true u and v. Ensure the lens holder keeps the lens upright.
Illuminated object not bright enough. In a well-lit room, the image on the screen may be faint and hard to focus. Darken the room and use a bright lamp behind the crosswire or wire gauze.
Object distance too close to f. When u is only slightly larger than f, the image distance v becomes very large and the image is dim. Readings at extreme values of u are less reliable. Avoid using u values within about 1.5f
10 | Common Exam Mistakes
Confusing real and virtual images. A converging lens produces a real image only when u>f. If the object is inside the focal length (u<f), the image is virtual, upright, and magnified --- it cannot be captured on a screen. This experiment only works with real images.
Sign convention errors. The O-Level syllabus uses the real-is-positive convention: u, v, and f are all positive for a converging lens forming a real image. Do not introduce negative signs unless the question explicitly uses a different sign convention.
Not using enough data points. Calculating f from a single pair of u and v is unreliable. You need at least 5--6 data points to draw a meaningful best-fit line and to identify anomalous readings.
Measuring from the wrong part of the lens. Always measure u and v from the centre (optical centre) of the lens, not from the edge of the lens or from the lens holder.
Plotting the graph with wrong axes. The standard plot is 1/v on the y-axis against 1/u on the x-axis. Swapping the axes changes which intercept gives 1/f. Stick to the convention stated in your practical instructions.
Forgetting to check for anomalies. After plotting, look for any point that sits far from the best-fit line. Investigate whether that reading had a large v (object near f) or a measurement error. Mark anomalous points and do not force the line through them.
Stating the focal length without units. Always include the unit (cm) when giving your final answer. A number without a unit is incomplete.
11 | Frequently Asked Questions
Why must the object distance be greater than the focal length?
When u>f, the converging lens produces a real, inverted image on the other side of the lens. This image can be caught on a screen and its position measured. When u<f, no real image forms --- the lens acts as a magnifying glass and produces a virtual image on the same side as the object. You cannot project a virtual image onto a screen, so you cannot measure v.
What happens when the object is at exactly 2f?
When u=2f, the image forms at v=2f as well. The image is the same size as the object magnification=1 and is real and inverted. This is a useful quick check: if you find a position where the image is the same size as the object, the focal length is half that distance.
Can I use a diverging lens for this experiment?
No. A diverging (concave) lens always produces a virtual image for a real object. Since the image cannot be projected onto a screen, you cannot use this direct method to measure its focal length. Diverging lenses require indirect methods (such as combining them with a known converging lens).
Why does the graph have a gradient of -1?
From the rearranged lens equation, 1/v=−1/u+1/f, the coefficient of 1/u is −1. This means that for every increase in 1/u, there is an equal decrease in 1/v. Checking that your graph gradient is close to −1 is a good way to verify your data --- a gradient significantly different from −1 suggests systematic measurement errors.
How accurate is the distant object method compared to the graph method?
The distant object method typically gives a value of f accurate to within about 5--10 %, depending on how far away the object is. The graph method, using six or more carefully measured data points, can achieve accuracy within 1--2 %. For an O-Level practical assessment, the graph method is expected because it demonstrates data collection, plotting, and analysis skills.
